NAG CL Interface
s17awc (airy_​ai_​deriv_​vector)

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1 Purpose

s17awc returns an array of values of the derivative of the Airy function Ai(x).

2 Specification

#include <nag.h>
void  s17awc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s17awc, nag_specfun_airy_ai_deriv_vector or nag_airy_ai_deriv_vector.

3 Description

s17awc evaluates an approximation to the derivative of the Airy function Ai(xi) for an array of arguments xi, for i=1,2,,n. It is based on a number of Chebyshev expansions.
For x<-5,
Ai(x)=-x4 [a(t)cosz+b(t)ζsinz] ,  
where z= π4+ζ, ζ= 23-x3 and a(t) and b(t) are expansions in variable t=-2 ( 5x) 3-1.
For -5x0,
Ai(x)=x2f(t)-g(t),  
where f and g are expansions in t=-2 ( x5) 3-1.
For 0<x<4.5,
Ai(x)=e-11x/8y(t),  
where y(t) is an expansion in t=4 ( x9)-1.
For 4.5x<9,
Ai(x)=e-5x/2v(t),  
where v(t) is an expansion in t=4 ( x9)-3.
For x9,
Ai(x) = x 4 e-z u(t) ,  
where z= 23x3 and u(t) is an expansion in t=2 ( 18z)-1.
For |x|< the square of the machine precision, the result is set directly to Ai(0). This both saves time and avoids possible intermediate underflows.
For large negative arguments, it becomes impossible to calculate a result for the oscillating function with any accuracy and so the function must fail. This occurs for x<- ( πε ) 4/7 , where ε is the machine precision.
For large positive arguments, where Ai decays in an essentially exponential manner, there is a danger of underflow so the function must fail.

4 References

NIST Digital Library of Mathematical Functions

5 Arguments

1: n Integer Input
On entry: n, the number of points.
Constraint: n0.
2: x[n] const double Input
On entry: the argument xi of the function, for i=1,2,,n.
3: f[n] double Output
On exit: Ai(xi), the function values.
4: ivalid[n] Integer Output
On exit: ivalid[i-1] contains the error code for xi, for i=1,2,,n.
ivalid[i-1]=0
No error.
ivalid[i-1]=1
xi is too large and positive. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_GT in s17ajc , as defined in the Users' Note for your implementation.
ivalid[i-1]=2
xi is too large and negative. f[i-1] contains zero. The threshold value is the same as for fail.code= NE_REAL_ARG_LT in s17ajc , as defined in the Users' Note for your implementation.
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of x was invalid.
Check ivalid for more information.

7 Accuracy

For negative arguments the function is oscillatory and hence absolute error is the appropriate measure. In the positive region the function is essentially exponential in character and here relative error is needed. The absolute error, E, and the relative error, ε, are related in principle to the relative error in the argument, δ, by
E |x2Ai(x)|δε | x2 Ai(x) Ai(x) |δ.  
In practice, approximate equality is the best that can be expected. When δ, ε or E is of the order of the machine precision, the errors in the result will be somewhat larger.
For small x, positive or negative, errors are strongly attenuated by the function and hence will be roughly bounded by the machine precision.
For moderate to large negative x, the error, like the function, is oscillatory; however, the amplitude of the error grows like
|x|7/4π.  
Therefore, it becomes impossible to calculate the function with any accuracy if |x|7/4> πδ .
For large positive x, the relative error amplification is considerable:
εδx3.  
However, very large arguments are not possible due to the danger of underflow. Thus in practice error amplification is limited.

8 Parallelism and Performance

s17awc is not threaded in any implementation.

9 Further Comments

None.

10 Example

This example reads values of x from a file, evaluates the function at each value of xi and prints the results.

10.1 Program Text

Program Text (s17awce.c)

10.2 Program Data

Program Data (s17awce.d)

10.3 Program Results

Program Results (s17awce.r)